| L(s) = 1 | − 2-s − 0.381·3-s + 4-s + 0.381·6-s − 3.85·7-s − 8-s − 2.85·9-s − 0.763·11-s − 0.381·12-s + 3.23·13-s + 3.85·14-s + 16-s − 5.23·17-s + 2.85·18-s − 2.76·19-s + 1.47·21-s + 0.763·22-s − 5.38·23-s + 0.381·24-s − 3.23·26-s + 2.23·27-s − 3.85·28-s − 3.61·29-s − 9.70·31-s − 32-s + 0.291·33-s + 5.23·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.220·3-s + 0.5·4-s + 0.155·6-s − 1.45·7-s − 0.353·8-s − 0.951·9-s − 0.230·11-s − 0.110·12-s + 0.897·13-s + 1.03·14-s + 0.250·16-s − 1.26·17-s + 0.672·18-s − 0.634·19-s + 0.321·21-s + 0.162·22-s − 1.12·23-s + 0.0779·24-s − 0.634·26-s + 0.430·27-s − 0.728·28-s − 0.671·29-s − 1.74·31-s − 0.176·32-s + 0.0507·33-s + 0.897·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 0.381T + 3T^{2} \) |
| 7 | \( 1 + 3.85T + 7T^{2} \) |
| 11 | \( 1 + 0.763T + 11T^{2} \) |
| 13 | \( 1 - 3.23T + 13T^{2} \) |
| 17 | \( 1 + 5.23T + 17T^{2} \) |
| 19 | \( 1 + 2.76T + 19T^{2} \) |
| 23 | \( 1 + 5.38T + 23T^{2} \) |
| 29 | \( 1 + 3.61T + 29T^{2} \) |
| 31 | \( 1 + 9.70T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 - 5.61T + 41T^{2} \) |
| 43 | \( 1 - 6.85T + 43T^{2} \) |
| 47 | \( 1 - 7.32T + 47T^{2} \) |
| 53 | \( 1 + 12.9T + 53T^{2} \) |
| 59 | \( 1 + 7.23T + 59T^{2} \) |
| 61 | \( 1 - 7.85T + 61T^{2} \) |
| 67 | \( 1 + 2.47T + 67T^{2} \) |
| 71 | \( 1 + 5.23T + 71T^{2} \) |
| 73 | \( 1 - 3.23T + 73T^{2} \) |
| 79 | \( 1 - 13.4T + 79T^{2} \) |
| 83 | \( 1 + 7.09T + 83T^{2} \) |
| 89 | \( 1 + 3.09T + 89T^{2} \) |
| 97 | \( 1 + 4.18T + 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.24278457073278389066229118257, −10.74171767110717545667155991047, −9.462149391786354292622844856013, −8.901093762585590430448340849961, −7.71593868234262445010192609927, −6.39206585478760235896483922085, −5.89636700991578969123487616027, −3.87892049051011056547608685235, −2.50203415052313896915087965507, 0,
2.50203415052313896915087965507, 3.87892049051011056547608685235, 5.89636700991578969123487616027, 6.39206585478760235896483922085, 7.71593868234262445010192609927, 8.901093762585590430448340849961, 9.462149391786354292622844856013, 10.74171767110717545667155991047, 11.24278457073278389066229118257
